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1-factor 2-valent vertices A-trail adjacent algorithm Beineke blocks breadth-first search chord circuit decomposition circuit double cover Class complete bipartite graphs complete graph composition operations Congressus Numerantium connected construct contains corresponding cutset cutvertex cutvertices cycle cyclic defined degree deleted denote depth-first search digraphs display embedding Erdos eulerian graph eulerian trail example exists factor families of graphs finite Frank Harary given graph G Graph Theory graphs are reconstructible harmonious coloring Hence hypomorphic hypomorphic mapping set incident infinite graphs integer isomorphic joined k-terminal recursive family k-trees labels Lemma Let G linear Mathematics matrices method multigraph non-reconstructable digraphs number of edges number of vertices obtained orbits pairs perfect matching permutation planar graphs plane drawing Pn x Pn problem procedure prove random graph reconstruction conjecture result series-parallel graphs signed graphs simple graph spanning tree Stockmeyer structure system of transitions tournaments triangle vertex